A063747 Sign of n-th coefficient of power series for 1/Gamma(1-x) where Gamma is the Gamma function.

1, -1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, -1, 1, 1

Offset

0,1

Comments

Computing this sequence was discussed on the seqfan mailing list in May 2023. Typical floating-point precision is not sufficient to compute more than a few terms of this sequence. Therefore, either a high precision needs to be set; e.g, a precision of round(n*log(n)/Pi) [see formula (3.21) in Fekih-Ahmed], or the computation must be done with an arbitrary precision method (e.g. interval arithmetic, FLINT, or the Java below). - Sean A. Irvine, May 21 2023

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Lazhar Fekih-Ahmed, On the Power Series Expansion of the Reciprocal Gamma Function, 2014.

Sean A. Irvine, Java program (github)

Example

1/Gamma(1-x) = 1 - gamma*x - 0.6558...*x^2 + 0.042...*x^3 + 0.1665...*x^4 + 0.0421...*x^5 - 0.0096..*x^6 +.... hence sequence begins 1,-1,-1,1,1,1,-1,...
[x^46] 1/Gamma(1-x) = -4.445829736550756882101590352124643637401436685748718288670...*10^-39, from which a(46)=-1. Sean A. Irvine, May 21 2023

Mathematica

precis = 200;
g[1, k_] := g[1, k] = EulerGamma^k/k!;
g[m_, k_] := g[m, k] = Zeta[m]^k/(k! m^k) // N[#, precis]&;
f[0, 0] = 1; f[n_ /; n > 0, 0] = 0;
f[n_, m_] := f[n, m] = Sum[g[m, k]*f[n - k m, m - 1],
  {k, 0, n/m}] // N[#, precis]&;
c[i_] := f[i, i]; b[0] = 1;
b[i_] := b[i] = -Sum[c[j]*b[i - j], {j, 1, i}];
a[n_] := Sign[b[n]];
Table[a[n], {n, 0, 74}] (* Jean-François Alcover, May 20 2023, after Brendan McKay on seqfan *)

Prog

(PARI) a(n)=sign(polcoeff(1/(gamma(1-x +O(x^(n+1)))), n))
(PARI) my(x='x+O('x^100)); apply(sign, Vec(1/(gamma(1-x)))) \\ Michel Marcus, May 19 2023

Crossrefs

Sequence in context: A209615 A242179 A319117 * A210245 A210247 A256175

Adjacent sequences: A063744 A063745 A063746 * A063748 A063749 A063750

Keyword

sign

Author

Benoit Cloitre, Jan 16 2004

Extensions

a(46) onward corrected by Sean A. Irvine, May 19 2023

Status

approved